author | wenzelm |
Fri, 27 Nov 2020 11:41:43 +0100 | |
changeset 72740 | 082200ee003d |
parent 66453 | cc19f7ca2ed6 |
permissions | -rw-r--r-- |
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(* Title: HOL/Corec_Examples/Tests/Small_Concrete.thy |
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Author: Aymeric Bouzy, Ecole polytechnique |
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Author: Jasmin Blanchette, Inria, LORIA, MPII |
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Copyright 2015, 2016 |
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Small concrete examples. |
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*) |
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section \<open>Small Concrete Examples\<close> |
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theory Small_Concrete |
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session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
wenzelm
parents:
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imports "HOL-Library.BNF_Corec" |
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begin |
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subsection \<open>Streams of Natural Numbers\<close> |
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codatatype natstream = S (head: nat) (tail: natstream) |
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corec (friend) incr_all where |
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"incr_all s = S (head s + 1) (incr_all (tail s))" |
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corec all_numbers where |
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"all_numbers = S 0 (incr_all all_numbers)" |
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corec all_numbers_efficient where |
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"all_numbers_efficient n = S n (all_numbers_efficient (n + 1))" |
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corec remove_multiples where |
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"remove_multiples n s = |
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(if (head s) mod n = 0 then |
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S (head (tail s)) (remove_multiples n (tail (tail s))) |
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else |
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S (head s) (remove_multiples n (tail s)))" |
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corec prime_numbers where |
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"prime_numbers known_primes = |
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(let next_prime = head (fold (%n s. remove_multiples n s) known_primes (tail (tail all_numbers))) in |
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S next_prime (prime_numbers (next_prime # known_primes)))" |
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term "prime_numbers []" |
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corec prime_numbers_more_efficient where |
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"prime_numbers_more_efficient n remaining_numbers = |
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(let remaining_numbers = remove_multiples n remaining_numbers in |
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S (head remaining_numbers) (prime_numbers_more_efficient (head remaining_numbers) remaining_numbers))" |
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term "prime_numbers_more_efficient 0 (tail (tail all_numbers))" |
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corec (friend) alternate where |
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"alternate s1 s2 = S (head s1) (S (head s2) (alternate (tail s1) (tail s2)))" |
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corec (friend) all_sums where |
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"all_sums s1 s2 = S (head s1 + head s2) (alternate (all_sums s1 (tail s2)) (all_sums (tail s1) s2))" |
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corec app_list where |
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"app_list s l = (case l of |
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[] \<Rightarrow> s |
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| a # r \<Rightarrow> S a (app_list s r))" |
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friend_of_corec app_list where |
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"app_list s l = (case l of |
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[] \<Rightarrow> (case s of S a b \<Rightarrow> S a b) |
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| a # r \<Rightarrow> S a (app_list s r))" |
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sorry |
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corec expand_with where |
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"expand_with f s = (let l = f (head s) in S (hd l) (app_list (expand_with f (tail s)) (tl l)))" |
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friend_of_corec expand_with where |
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"expand_with f s = (let l = f (head s) in S (hd l) (app_list (expand_with f (tail s)) (tl l)))" |
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sorry |
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corec iterations where |
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"iterations f a = S a (iterations f (f a))" |
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corec exponential_iterations where |
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"exponential_iterations f a = S (f a) (exponential_iterations (f o f) a)" |
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corec (friend) alternate_list where |
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"alternate_list l = (let heads = (map head l) in S (hd heads) (app_list (alternate_list (map tail l)) (tl heads)))" |
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corec switch_one_two0 where |
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"switch_one_two0 f a s = (case s of |
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S b r \<Rightarrow> S b (S a (f r)))" |
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corec switch_one_two where |
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"switch_one_two s = (case s of |
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S a (S b r) \<Rightarrow> S b (S a (switch_one_two r)))" |
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corec fibonacci where |
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"fibonacci n m = S m (fibonacci (n + m) n)" |
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corec sequence2 where |
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"sequence2 f u1 u0 = S u0 (sequence2 f (f u1 u0) u1)" |
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corec (friend) alternate_with_function where |
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"alternate_with_function f s = |
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(let f_head_s = f (head s) in S (head f_head_s) (alternate (tail f_head_s) (alternate_with_function f (tail s))))" |
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corec h where |
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"h l s = (case l of |
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[] \<Rightarrow> s |
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| (S a s') # r \<Rightarrow> S a (alternate s (h r s')))" |
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friend_of_corec h where |
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"h l s = (case l of |
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[] \<Rightarrow> (case s of S a b \<Rightarrow> S a b) |
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| (S a s') # r \<Rightarrow> S a (alternate s (h r s')))" |
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sorry |
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corec z where |
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"z = S 0 (S 0 z)" |
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lemma "\<And>x. x = S 0 (S 0 x) \<Longrightarrow> x = z" |
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apply corec_unique |
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apply (rule z.code) |
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done |
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corec enum where |
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"enum m = S m (enum (m + 1))" |
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lemma "(\<And>m. f m = S m (f (m + 1))) \<Longrightarrow> f m = enum m" |
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apply corec_unique |
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apply (rule enum.code) |
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done |
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lemma "(\<forall>m. f m = S m (f (m + 1))) \<Longrightarrow> f m = enum m" |
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apply corec_unique |
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apply (rule enum.code) |
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done |
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subsection \<open>Lazy Lists of Natural Numbers\<close> |
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codatatype llist = LNil | LCons nat llist |
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corec h1 where |
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"h1 x = (if x = 1 then |
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LNil |
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else |
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let x = if x mod 2 = 0 then x div 2 else 3 * x + 1 in |
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LCons x (h1 x))" |
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corec h3 where |
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"h3 s = (case s of |
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LNil \<Rightarrow> LNil |
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| LCons x r \<Rightarrow> LCons x (h3 r))" |
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corec fold_map where |
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"fold_map f a s = (let v = f a (head s) in S v (fold_map f v (tail s)))" |
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friend_of_corec fold_map where |
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"fold_map f a s = (let v = f a (head s) in S v (fold_map f v (tail s)))" |
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apply (rule fold_map.code) |
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sorry |
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subsection \<open>Coinductive Natural Numbers\<close> |
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codatatype conat = CoZero | CoSuc conat |
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corec sum where |
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"sum x y = (case x of |
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CoZero \<Rightarrow> y |
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| CoSuc x \<Rightarrow> CoSuc (sum x y))" |
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friend_of_corec sum where |
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"sum x y = (case x of |
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CoZero \<Rightarrow> (case y of CoZero \<Rightarrow> CoZero | CoSuc y \<Rightarrow> CoSuc y) |
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| CoSuc x \<Rightarrow> CoSuc (sum x y))" |
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sorry |
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corec (friend) prod where |
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"prod x y = (case (x, y) of |
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(CoZero, _) \<Rightarrow> CoZero |
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| (_, CoZero) \<Rightarrow> CoZero |
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| (CoSuc x, CoSuc y) \<Rightarrow> CoSuc (sum (prod x y) (sum x y)))" |
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end |